articles - With or without 'the' in "in (the) present participle form"?

Consider the two following extracts:

[...], which precedes the main verb in the present participle form, working.
_{–The Teacher's Grammar of English with Answers: A Course Book and Reference Guide}

If you use the verb dye in present participle form, the e must be retained to avoid confusion: dyeing.
_{–Write in Style: A guide to good English}

^{(Emphasis mine)}

Both sources appear to be about good English. One is for teachers; the other is for good writing. One of them uses "the present participle form" (with the), whereas the other uses "present participle form" (without the).

Does the really matter at all in the extracts above? Is one of them more correct (or considered good usage) than the other? Or is it the case that both of them are correct but hint at different things?

Answer

He's in first grade and learning his ABCs.
He's in the first grade and learning his ABCs.

There is no meaningful (i.e. deliberately evoked and clearly perceived) difference. Both are acceptable and grammatical, and both could appear in the same idiolect in the same register.

Stages, forms, tiers, phases, grades, levels etc are perceived differently than spoons, cups, pencils, stones. The former can be projected as an isolated abstraction, or alternatively as belonging to an ordered sequence. When it's the latter, the definite article is used.

But...

^OK Many of the passengers up in first class survived.

^{not OK} Many of the passengers up in the first class survived.

^OK Many students start thinking of college in junior year.

^OK Many students start thinking of college in the junior year.

Unlike "first grade" and "the first grade", with "junior year" and "the junior year" there is a nuanced difference, one of register, the latter being the preference of bureaucrats. Someone who had gone only as far as elementary school could easily use both "in first grade" and "in the first grade", but someone who uses "in the junior year" is either a professional in the education biz or rubs elbows with such folks on a regular basis.